The Reeh-Schlieder theorem says that every target vector can be approximated from the vacuum by an operator localized in an arbitrarily small spacetime region, but it gives no quantitative cost for doing so. This note isolates a standard Tomita-Takesaki estimate as a model-independent preparation bound. Targets with deeply negative modular energy require large local operators. After rescaling such an operator to a physical contraction, the same estimate becomes a lower bound on postselection overhead. In geometries where the modular Hamiltonian is known, the bound becomes explicit. Bisognano-Wichmann turns it into a boost energy statement for wedges, and the Casini-Huerta-Myers formula gives a stress-tensor version for bounded regions of conformal field theories. Local unitaries can only reach states of nonnegative modular energy. Negative modular sectors require nonunitary or postselected outcomes, giving a preparation cost bound that complements vacuum embezzlement in type III local algebras.
The Reeh-Schlieder theorem says that every target vector can be approximated from the vacuum by an operator localized in an arbitrarily small spacetime region, but it gives no quantitative cost for doing so. This note isolates a standard Tomita-Takesaki estimate as a model-independent preparation bound. Targets with deeply negative modular energy require large local operators. After rescaling such an operator to a physical contraction, the same estimate becomes a lower bound on postselection overhead. In geometries where the modular Hamiltonian is known, the bound becomes explicit. Bisognano-Wichmann turns it into a boost energy statement for wedges, and the Casini-Huerta-Myers formula gives a stress-tensor version for bounded regions of conformal field theories. Local unitaries can only reach states of nonnegative modular energy. Negative modular sectors require nonunitary or postselected outcomes, giving a preparation cost bound that complements vacuum embezzlement in type III local algebras.
The general scope of this PhD thesis is to establish relations between Tomita-Takesaki modular theory, in particular modular Hamiltonians and relative modular Hamiltonians, and physically better understood quantities, such as the stress-energy tensor, energy densities or other modular data in cases where it is known explicitly. In Chapter 1, we will present the basics of operators algebras, Tomita-Takesaki theory, symplectic spaces, and link these concepts to discuss quasi-equivalence of Gaussian states. In Chapter 2, we will discuss algebraic QFTs both in the framework of the Haag-Kastler axioms and of locally covariant QFT, to then specialize to scalar fields and present the stress-energy tensor and the Hadamard condition. In Chapter 3, focussing on Gaussian states of the Weyl-CCR algebra, we study relations between quasi-equivalence of their GNS-representations, differences of the total energy (on some Cauchy surface), and differences between functions of the modular Hamiltonians. Following qualitative relations by Araki and Yamagami, Verch and Longo, we investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. These results will make use of inequalities between p-norms of operators, which we will discuss in Appendix C. In Chapter 4, generalizing the picture beyond Gaussian states, we exploit locality properties of general algebraic QFTs to construct a scheme that allows to estimate relative modular Hamiltonians for suitable pairs of states on some local algebra in terms of modular Hamiltonians of a reference state on bigger or smaller algebras, which might be better understood. For pairs of states susceptible to this scheme, there exist unitaries linking them to the reference state which do not enable superluminal signalling in the sense of Sorkin's paradox. Such unitaries may not however compose the whole class of non-signalling operators, raising the question whether the existence of a non-signalling unitary is a sufficient condition to show the applicability of our estimate. To investigate the strength of our estimates we then consider coherent states for CCR systems, focussing particularly on free scalar fields, where the difference between the upper and lower bounds can be made arbitrarily small, recovering an exact result by squeezing. This yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian has not been computed exactly and corroborates the strength of our general estimates. For these results, we will make use of several commutation properties of operators on Fock spaces, which we will present in Appendix A. Results about operator monotone functions will be used throughout this thesis and are explained in Appendix B.
The Tomita-Takesaki modular theory is employed to discuss the Bell-CHSH inequality in wedge regions. By using the Bisognano-Wichmann results, the construction of a set of wedge localized vectors in the one-particle Hilbert space of a relativistic massive scalar field in 1+1 dimensions is devised to establish whether violations of the Bell-CHSH inequality might occur for different choices of Bell's operators. In particular, the construction of the wedge localized vectors employed in the seminal work by Summers-Werner is scrutinized and applied to Weyl and other operators. We also outline a possible path towards the saturation of Tsirelson's bound.
We ask in which states, and for which local algebras, Galilean quantum fields have the Reeh-Schlieder property. The nets are generated by time-zero canonical fields on spatial regions and by the dynamics. The Bargmann mass over the charge of the field is a number operator for the time-zero fields, bounded from below by positivity of the energy and boost covariance, and Chaiken's theorem gives Galilean Fock rigidity: these fields form a direct sum of Fock representations, and the time-zero annihilation fields annihilate every translation-invariant vector. If the vacuum is the only vector invariant under spatial translations (not implied by the usual uniqueness axiom), the vacuum sector of the observable net is one-dimensional. No vector of finite mass is cyclic or separating for an algebra of a spatial region at fixed time. In the vacuum of Bose gases with bounded or Coulomb-type pair potentials (with Buchholz's renormalised dynamics if not H-stable), the fields of any region over any time interval generate all bounded operators. By contrast, a KMS state is cyclic and separating for the algebra of every region B×I with temporal extent whenever the spatial algebras at all times generate the global algebra, as for free gases. For the free Schrödinger field in the vacuum, the thermal states of Bose and Fermi gases and the Fermi sea, fixed-time field algebras are type I factors, for which thermal vectors and the Fermi sea are separating but not cyclic, and every field algebra with temporal extent is the global one: type I in pure states, type III1 in thermal states, with the grand-canonical dynamics as modular group. Galilean vacua are thus never separating for local algebras. For the free field the type of the algebras of regions with temporal extent depends on the state, and what is specifically relativistic is the locality of type III1.
Abstract The Tomita–Takesaki modular theory is employed to discuss the Bell-CHSH inequality in wedge regions. By using the Bisognano–Wichmann results, the construction of a set of wedge localized vectors in the one-particle Hilbert space of a relativistic massive scalar field in 1+1 1 + 1 dimensions is devised to establish whether violations of the Bell-CHSH inequality might occur for different choices of Bell’s operators. In particular, the construction of the wedge localized vectors employed in the seminal work by Summers–Werner is scrutinized and applied to Weyl and other operators. We also outline a possible path towards the saturation of Tsirelson’s bound.
In this work we study the Bell-CHSH inequality in the vacuum of the free massive Majorana field in 1+1 dimensions, taking as the dichotomic observables of the two parties even bilinears of smeared fields localized in the right and in the left Rindler wedges. The test functions are generated from seeds in the modular-frequency variable through the Tomita-Takesaki operator of the right wedge and, on the side of the left wedge, through its adjoint multiplied by i, which accounts for twisted duality. For seeds supported at positive modular frequency the one-particle overlaps between the two parties do not depend on the modular weights, and we obtain the Bell parameter in closed form. When the seeds are phase rotations of a single compact bump, the bilinears generate a copy of the Pauli algebra in each wedge, and we compute exactly the correlation matrix of the vacuum restricted to the two copies. The optimal settings violate the classical bound up to spectral widths more than twice as large as those reached with planar settings. We also show that the Tsirelson bound is never attained in the wedge vacuum for any choice of dichotomic observables, and that approaching it forces the modular spectral weight of the Bell vectors to concentrate at the spectral point λ=1 of the modular operator. This establishes in a precise way how the even sector of the free Majorana field realizes the Summers-Werner mechanism.
The vacuum modular Hamiltonian K of the Rindler wedge in any relativistic quantum field theory is given by the boost generator. Here we investigate the modular Hamiltonian for more general half-spaces which are bounded by an arbitrary smooth cut of a null plane. We derive a formula for the second derivative of the modular Hamiltonian with respect to the coordinates of the cut which schematically reads ${K}^{\ensuremath{'}\ensuremath{'}}={T}_{vv}$. This formula can be integrated twice to obtain a simple expression for the modular Hamiltonian. The result naturally generalizes the standard expression for the Rindler modular Hamiltonian to this larger class of regions. Our primary assumptions are the quantum null energy condition---an inequality between the second derivative of the von Neumann entropy of a region and the stress tensor---and its saturation in the vacuum for these regions. We discuss the validity of these assumptions in free theories and holographic theories to all orders in 1/N.